Abstract
An element \(x \in X\) is called a fixed point of the mapping \(f:X \rightarrow X\) if \(f(x) = x\). Many problems, looking rather dissimilar at a first glance, from various domains of mathematics, can be reduced to the search for fixed points of appropriate mappings. For this reason each of the theorems on existence of fixed points discussed in the present chapter has numerous and often very elegant applications. Below we present three classical fix-point theorems: the Banach’s theorem on contractive mappings, the Schauder’s principle (which are applied for demonstrations of the Picard and Peano theorems on the existence of a solution to the Cauchy problem for differential equations and to the Lomonosov invariant subspace theorem), and the Kakutani’s theorem about common fixed points of a family of isometries, with application to the existence of a Haar measure on a compact topological group.
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Notes
- 1.
The proof mentioned here was proposed by B. Knaster, S. Mazurkiewicz, and K. Kuratowski, so that the treatment in [25] belongs to one of these authors. Generally, we find it very useful to refer beginners to textbooks written not simply by pedagogues who are familiar with and know well how to treat the material, but by people who contributed in an essential manner to the creation and development of the corresponding fields of mathematics. In such texts the reader has the opportunity to make acquaintance not only with the results, but also — even more importantly — with the ways people who demonstrated the fruitfulness of their approach to mathematical research think.
- 2.
That is, \(\varphi _s\geqslant 0\), \(\sum _{s\in J} \varphi _s \equiv 1\), and the support of the function \(\varphi _s\) lies in the corresponding set \(V_s\), see Subsection 15.1.3.
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Kadets, V. (2018). Fixed Point Theorems and Applications. In: A Course in Functional Analysis and Measure Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-92004-7_15
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DOI: https://doi.org/10.1007/978-3-319-92004-7_15
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